I'm running this plugin on two sites that I manage: http://theinfertilityvoice.com and http://www.resolvenewengland.org. In both instances, you can see that the "Select Tag" dropdown menu does not match the fonts being used in either theme (running Genesis Scribble on the former, Genesis Lifestyle on the latter).

I imagine this is a fix somewhere in the plugin files, but not sure where to start. Any help on how to get the dropdown menus to match the default fonts on my sites would be much appreciated!

The fix for this would be made in your site's CSS, rather than in my plugin. That said, support for styling dropdowns is largely operating system and browser dependent. This means that while you may be able to style the dropdown in some browsers, it likely won't result in a consistent visual looks for all visitors.

The taxonomy_dropdown_widget_dropdown class appears on every dropdown added by my plugin, so you can attempt to style them using that class.